What can one say about the integrability of solutions of the time-independent Schroedinger equation $H\psi=E\psi$, where $H=p^2/2m +V(x)$, with $V(x)=W(x)^2$?

a) For bound states, $\psi$ is square integrable. Is the same true for $p\psi$? For $W(x)\psi$?

b) For scattering states? Here $\psi$ is not square integrable. What are the requirements on $\phi$ such that $\langle\phi|\psi\rangle$ exists? Is the square integrability of $H\psi$ enough? Or that of $p\psi$ and $W(x)\psi$?

Where can I find such questions discussed?

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